On intersections of primary subgroups pairs in finite group with socle $\Omega_{2n}^+(2^m)$

In theorem 1 for $Soc(G) = \Omega_{2n}^+(2)$, $n \ge 3$ and $S \in Syl_2(G)$ subgroup $min_G(S,S) = \langle S \bigcap S^g | |S \bigcap S^g| is\ minimal \rangle$ is constructed. In theorem 2 it is proved that if $Soc(G) = \Omega_{2n}^+(2^m)$ and for primary subgroups $A$ and $B$ we have $min_G(A,B) \ne 1$, then $m=1$, we can assume that $A$ and $B$ are subgroups of $S \in Syl_2(G)$, $|G:Soc(G)|=2$, involution from $G-Soc(G)$ induces the graph automorphism on $Soc(G)$ and $min_G(S,S)\subseteq A\cap B$.